91Ó°ÊÓ

A system of linear equations is a collection of one or more linear equations involving the same set of variables. In our example, we have three equations with three variables: \(x\), \(y\), and \(z\). These equations represent lines in three-dimensional space:

• \(4x - 2y + 3z = -2\)
• \(2x + 2y + 5z = 16\)
• \(8x - 5y - 2z = 4\)

Our goal is to find the values of \(x\), \(y\), and \(z\) that satisfy all these equations simultaneously. This is essential in many fields like physics for solving force balance problems or in economics for finding equilibrium conditions.

The determinant is a special number that can be calculated from a square matrix. It provides important properties about the matrix, such as whether it has an inverse. In this process, the determinant of matrix \(A\) plays a crucial role.
For our matrix \(A\):
\[A = \begin{bmatrix} 4 & -2 & 3 \2 & 2 & 5 \8 & -5 & -2 \end{bmatrix}\]The determinant is calculated using specific expansion methods, often involving minors and cofactors. Fortunately, if the determinant is not zero, as is the case here (\(-12\)), it assures us that the matrix is invertible.
The determinant tells us crucial things about the system of equations, primarily that there exists a unique solution, as opposed to having no solutions or infinitely many.

Matrix multiplication is used when solving systems of equations through matrices, especially when involving inverse matrices. To find the solution of the equation \(AX = B\), we first need the product of the inverse matrix \(A^{-1}\) and the matrix of constants \(B\).
Given:
\[A^{-1} = \begin{bmatrix} -1/6 & 4/3 & 0 \-1/6 & 5/6 & 1/6 \2/3 & -8/3 & 1/6 \end{bmatrix}\]
And:
\[B = \begin{bmatrix} -2 \16 \4 \end{bmatrix}\]
The product \(A^{-1}B\) is computed by performing the dot product of rows from \(A^{-1}\) with the column from \(B\). This operation results in the matrix \(X\) containing our solutions.

The solution to the system of linear equations is derived from the matrix multiplication of the inverse of the coefficients matrix and the constants matrix. The product yields:
\[X = A^{-1}B = \begin{bmatrix} 8 \0 \-4 \end{bmatrix}\]
This indicates that the values \(x = 8\), \(y = 0\), and \(z = -4\) satisfy all equations simultaneously. We verify this by substituting these values back into the original equations, ensuring each equation holds true. Solving systems like these helps understand real-world phenomena, modeling everything from engineering problems to financial scenarios.